{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:38:14Z","timestamp":1753889894719,"version":"3.41.2"},"reference-count":1,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2015,9,11]],"date-time":"2015-09-11T00:00:00Z","timestamp":1441929600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>The quantified constraint satisfaction problem $\\mathrm{QCSP}(\\mathcal{A})$\nis the problem to decide whether a positive Horn sentence, involving nothing\nmore than the two quantifiers and conjunction, is true on some fixed structure\n$\\mathcal{A}$. We study two containment problems related to the QCSP. Firstly,\nwe give a combinatorial condition on finite structures $\\mathcal{A}$ and\n$\\mathcal{B}$ that is necessary and sufficient to render\n$\\mathrm{QCSP}(\\mathcal{A}) \\subseteq \\mathrm{QCSP}(\\mathcal{B})$. We prove\nthat $\\mathrm{QCSP}(\\mathcal{A}) \\subseteq \\mathrm{QCSP}(\\mathcal{B})$, that is\nall sentences of positive Horn logic true on $\\mathcal{A}$ are true on\n$\\mathcal{B}$, iff there is a surjective homomorphism from\n$\\mathcal{A}^{|A|^{|B|}}$ to $\\mathcal{B}$. This can be seen as improving an\nold result of Keisler that shows the former equivalent to there being a\nsurjective homomorphism from $\\mathcal{A}^\\omega$ to $\\mathcal{B}$. We note\nthat this condition is already necessary to guarantee containment of the\n$\\Pi_2$ restriction of the QCSP, that is $\\Pi_2$-$\\mathrm{CSP}(\\mathcal{A})\n\\subseteq \\Pi_2$-$\\mathrm{CSP}(\\mathcal{B})$. The exponent's bound of\n${|A|^{|B|}}$ places the decision procedure for the model containment problem\nin non-deterministic double-exponential time complexity. We further show the\nexponent's bound $|A|^{|B|}$ to be close to tight by giving a sequence of\nstructures $\\mathcal{A}$ together with a fixed $\\mathcal{B}$, $|B|=2$, such\nthat there is a surjective homomorphism from $\\mathcal{A}^r$ to $\\mathcal{B}$\nonly when $r \\geq |A|$. Secondly, we prove that the entailment problem for\npositive Horn fragment of first-order logic is decidable. That is, given two\nsentences $\\varphi$ and $\\psi$ of positive Horn, we give an algorithm that\ndetermines whether $\\varphi \\rightarrow \\psi$ is true in all structures\n(models). Our result is in some sense tight, since we show that the entailment\nproblem for positive first-order logic (i.e. positive Horn plus disjunction) is\nundecidable. In the final part of the paper we ponder a notion of Q-core that\nis some canonical representative among the class of templates that engender the\nsame QCSP. Although the Q-core is not as well-behaved as its better known\ncousin the core, we demonstrate that it is still a useful notion in the realm\nof QCSP complexity classifications.<\/jats:p>","DOI":"10.2168\/lmcs-11(3:9)2015","type":"journal-article","created":{"date-parts":[[2016,11,21]],"date-time":"2016-11-21T13:46:02Z","timestamp":1479735962000},"source":"Crossref","is-referenced-by-count":3,"title":["Quantified Constraints and Containment Problems"],"prefix":"10.46298","volume":"Volume 11, Issue 3","author":[{"given":"Barnaby D.","family":"Martin","sequence":"first","affiliation":[]},{"given":"Hubie","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Florent R.","family":"Madelaine","sequence":"additional","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2015,9,11]]},"reference":[{"key":"1007:not-found"}],"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/lmcs.episciences.org\/1585\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/lmcs.episciences.org\/1585\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,11]],"date-time":"2023-04-11T20:07:12Z","timestamp":1681243632000},"score":1,"resource":{"primary":{"URL":"https:\/\/lmcs.episciences.org\/1585"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,9,11]]},"references-count":1,"URL":"https:\/\/doi.org\/10.2168\/lmcs-11(3:9)2015","relation":{"is-same-as":[{"id-type":"arxiv","id":"1310.1016","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1310.1016","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"type":"electronic","value":"1860-5974"}],"subject":[],"published":{"date-parts":[[2015,9,11]]},"article-number":"1585"}}